Associativity? Can this be applied here?

Question

As the Associativity law says that (A ∧ B) ∧ C ≡ (A ∧ C) ∧ B, can I do something like this?

(A ∧ ¬B) ∨ (B ∧ ¬A) ≡ (A ∧ ¬A) ∨ (B ∧ ¬B)

I am new with logic and I still don't get this basic principles..

Answer 1

Associativity is a property that applies when you have several operators all the same. So $$ (A \wedge \lnot B) \wedge (B \wedge \lnot A) \equiv (A \wedge \lnot A) \wedge (B \wedge \lnot B). $$ We can apply associativity there because every operator is $\wedge$. (We also have to apply commutativity to get this result.) But $\wedge$ and $\vee$ are different operators, so they don't associate: $$ (A \wedge \lnot B) \vee (B \wedge \lnot A) \not\equiv (A \wedge \lnot A) \vee (B \wedge \lnot B). \tag1 $$ We can see the two sides are not necessarily the same because there is a model that satisfies $A\wedge\lnot B$, therefore it satisfies the left-hand side of $(1)$, but the right-hand side of $(1)$ cannot be satisfied.